Group action invariants
Degree $n$: | $16$ | |
Transitive number $t$: | $50$ | |
Group: | $SD_{16}:C_2$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $3$ | |
$|\Aut(F/K)|$: | $8$ | |
Generators: | (1,8,2,7)(3,6,4,5)(9,12,10,11)(13,15,14,16), (9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,11)(4,12)(5,15)(6,16)(7,10)(8,9) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$
Low degree siblings
16T32, 32T18Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 9, 3,16, 2,10, 4,15)( 5,11, 7,13, 6,12, 8,14)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 9, 4,15, 2,10, 3,16)( 5,11, 8,14, 6,12, 7,13)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,10, 6, 9)( 7,16, 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,16)( 8,15)$ |
Group invariants
Order: | $32=2^{5}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [32, 44] |
Character table: |
2 5 4 5 4 4 3 3 3 3 3 3 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 2P 1a 1a 1a 2b 2b 2b 2b 4b 4b 2b 1a 3P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 5P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 7P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 -1 1 X.3 1 -1 1 -1 1 -1 1 1 -1 1 -1 X.4 1 -1 1 -1 1 1 -1 -1 1 1 -1 X.5 1 -1 1 -1 1 1 -1 1 -1 -1 1 X.6 1 1 1 1 1 -1 -1 -1 -1 1 1 X.7 1 1 1 1 1 -1 -1 1 1 -1 -1 X.8 1 1 1 1 1 1 1 -1 -1 -1 -1 X.9 2 2 2 -2 -2 . . . . . . X.10 2 -2 2 2 -2 . . . . . . X.11 4 . -4 . . . . . . . . |